Representation of an m-by-n matrix. It is required that
the linear operator can produce Ax and A^Tx.

b : (m,) ndarray

Right-hand side vector b.

damp : float

Damping coefficient.

atol, btol : float, optional

Stopping tolerances. If both are 1.0e-9 (say), the final
residual norm should be accurate to about 9 digits. (The
final x will usually have fewer correct digits, depending on
cond(A) and the size of damp.)

conlim : float, optional

Another stopping tolerance. lsqr terminates if an estimate of
cond(A) exceeds conlim. For compatible systems Ax=b, conlim could be as large as 1.0e+12 (say). For
least-squares problems, conlim should be less than 1.0e+8.
Maximum precision can be obtained by setting atol=btol=conlim=zero, but the number of iterations may then be
excessive.

If calc_var is True, estimates all diagonals of
(A'A)^{-1} (if damp==0) or more generally (A'A+damp^2*I)^{-1}. This is well defined if A has full column
rank or damp>0. (Not sure what var means if rank(A)<n and damp=0.)

Notes

LSQR uses an iterative method to approximate the solution. The
number of iterations required to reach a certain accuracy depends
strongly on the scaling of the problem. Poor scaling of the rows
or columns of A should therefore be avoided where possible.

For example, in problem 1 the solution is unaltered by
row-scaling. If a row of A is very small or large compared to
the other rows of A, the corresponding row of ( A b ) should be
scaled up or down.

In problems 1 and 2, the solution x is easily recovered
following column-scaling. Unless better information is known,
the nonzero columns of A should be scaled so that they all have
the same Euclidean norm (e.g., 1.0).

In problem 3, there is no freedom to re-scale if damp is
nonzero. However, the value of damp should be assigned only
after attention has been paid to the scaling of A.

The parameter damp is intended to help regularize
ill-conditioned systems, by preventing the true solution from
being very large. Another aid to regularization is provided by
the parameter acond, which may be used to terminate iterations
before the computed solution becomes very large.

If some initial estimate x0 is known and if damp==0,
one could proceed as follows:

Compute a residual vector r0=b-A*x0.

Use LSQR to solve the system A*dx=r0.

Add the correction dx to obtain a final solution x=x0+dx.

This requires that x0 be available before and after the call
to LSQR. To judge the benefits, suppose LSQR takes k1 iterations
to solve A*x = b and k2 iterations to solve A*dx = r0.
If x0 is “good”, norm(r0) will be smaller than norm(b).
If the same stopping tolerances atol and btol are used for each
system, k1 and k2 will be similar, but the final solution x0 + dx
should be more accurate. The only way to reduce the total work
is to use a larger stopping tolerance for the second system.
If some value btol is suitable for A*x = b, the larger value
btol*norm(b)/norm(r0) should be suitable for A*dx = r0.

Preconditioning is another way to reduce the number of iterations.
If it is possible to solve a related system M*x=b
efficiently, where M approximates A in some helpful way (e.g. M -
A has low rank or its elements are small relative to those of A),
LSQR may converge more rapidly on the system A*M(inverse)*z=b, after which x can be recovered by solving M*x = z.

If A is symmetric, LSQR should not be used!

Alternatives are the symmetric conjugate-gradient method (cg)
and/or SYMMLQ. SYMMLQ is an implementation of symmetric cg that
applies to any symmetric A and will converge more rapidly than
LSQR. If A is positive definite, there are other implementations
of symmetric cg that require slightly less work per iteration than
SYMMLQ (but will take the same number of iterations).